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CHAPTER I

ELEMENTS OF THE THEORY OF SETS

We do not try in this chapter to put set theory on an ^d^iatic basis;
this can however be done, and we refer the interested reader to Kelley [15]
and Bourbaki [3] for a complete axiomatic description. Statements appearing
in this chapter and which are not accompanied by a proof or a definition may
be considered as axioms connecting undefined terms.

The chapter starts with some elementary definitions and formulas about
sets, subsets and product sets (Sections 1.1 to 1.3); the bulk of the chapter is
devoted to the fuyod^mei^l notion^ of mapping, which is the modern ex-
tension of the classical concept of a (numerical) function of one or several
numerical "variables." Two points related to this concept^ deserve^some
comment:

1. The all-important (and characteristic) property of a mapping is that it
associates to any "value" of the variable a single element; in other words,
there is no such thing as a " multiple-valued " function, despite many books
to the contrary. It is of course perfectly legitimate to define a mapping whose
values are subsets of a given set, which may have more than one element;
but such definitions are in practice useless (at least in elementary analysis),
because it is impossible to define in a sensible way algebraic operations on the
"values" of such functions. We return to this question in Chapter IX.

2. The student should as soon as possible become familiar with the idea
that a function/is a single object, which may itself " vary " and is in general
to be thought of as a "point" in a large "functional space"; indeed, it may
be said that one of the main differences between the classical and the modern
concepts of analysis is that, in classical mathematics, when one writes f(x)₉/is visualized as "fixed" and x as "variable," whereas nowadays bothf



	CHAPTER I ELEMENTS OF THE THEORY OF SETS

	We do not try in this chapter to put set theory on an ^d^iatic basis; this can however be done, and we refer the interested reader to Kelley [15] and Bourbaki [3] for a complete axiomatic description. Statements appearing in this chapter and which are not accompanied by a proof or a definition may be considered as axioms connecting undefined terms. The chapter starts with some elementary definitions and formulas about sets, subsets and product sets (Sections 1.1 to 1.3); the bulk of the chapter is devoted to the fuyod^mei^l notion^ of mapping, which is the modern ex- tension of the classical concept of a (numerical) function of one or several numerical "variables." Two points related to this concept^ deserve^some comment: 1. The all-important (and characteristic) property of a mapping is that it associates to any "value" of the variable a single element; in other words, there is no such thing as a " multiple-valued " function, despite many books to the contrary. It is of course perfectly legitimate to define a mapping whose values are subsets of a given set, which may have more than one element; but such definitions are in practice useless (at least in elementary analysis), because it is impossible to define in a sensible way algebraic operations on the "values" of such functions. We return to this question in Chapter IX. 2. The student should as soon as possible become familiar with the idea that a function/is a single object, which may itself " vary " and is in general to be thought of as a "point" in a large "functional space"; indeed, it may be said that one of the main differences between the classical and the modern concepts of analysis is that, in classical mathematics, when one writes f(x)₉/is visualized as "fixed" and x as "variable," whereas nowadays bothf